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Material Type: Notes; Class: SYSTEMS ANALYSIS TECHNIQUES; Subject: Electrical & Comp. Sys. Engr.; University: Rensselaer Polytechnic Institute; Term: Fall 2004;
Typology: Study notes
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and Observability
. (^) define and explain the concepts of
. (^) explain the importance of complete controllability ana complete observaoility and relate them to full-state feedback controllers and observers. . (^) determine if a discrete-time system is
Ref: Sections 5.6, 5. 7. Fall 2004
General Concepts
I II
C1" I F 2n C2"1^ F
Controllability
. Each loop has a characteristic frequency (or mode) of e -t.
-+
t y
Draw a block diagram for
0 0
0 1 x(t) + lu(t) 1 0
-1 1
y(t) = [1 0 1 0 ]x(t)
Which states are observable? which are
controllable? Which states are part of the transfer function? What is the transfer function?
Complete Controllability (CC)
A system is CC iff for every possible x(tI) :3 a u(t), 0 < t < tl 3 the system with zero initial conditions can be forced to x(tI).
Alternate Definitions,
Del
Wh~t does this definition imply about the
Full-State Feedback and Observers
Full-state feedback implies that you can measure all of the states.
What if all the states cannot be measured?
knowledge of yet) and u(t).
Controllability Tests for Discrete-Time Systems
Find a control to take the system from an arbitrary x(O)to 0 in k steps.
x(k) = Akx(O)+ Ak-l Bu(O)
Since x(k) is the origin,
Now solve for u(O), u(l), ...
Test this single-input, single-output system for complete observability,
1 0 1 x(k+ 1) = I x(k) + lu(k) -1/2 1/2 -
y(k) = [5 1] x(k) (^) Ans. system is CO
Test this two-input system for Complete Controllability.
.5 .5 0 3 1
x(k+ 1) = 0 1 0 x(k) + 2 0 u(k) .83 -2.16 -.33 -1 1 Ans. system is not CC
3 1 2.5 .5 2.25.
8=202020c
-1 1 -1.5 .5 -1.75.
